scholarly journals Sumsets Contained in Sets of Upper Banach Density 1

2017 ◽  
pp. 239-248
Author(s):  
Melvyn B. Nathanson
Keyword(s):  
Banach Density

Sets of large values of correlation functions for polynomial cubic configurations

2016 ◽  
Vol 38 (2) ◽  
pp. 499-522 ◽  
Author(s):  
V. BERGELSON ◽  
A. LEIBMAN
Keyword(s):  
Correlation Functions ◽  
Banach Density ◽  
Image Position

We prove that for any set$E\subseteq \mathbb{Z}$with upper Banach density$d^{\ast }(E)>0$, the set ‘of cubic configurations’ in$E$is large in the following sense: for any$k\in \mathbb{N}$and any$\unicode[STIX]{x1D700}>0$, the set$$\begin{eqnarray}\displaystyle \biggl\{(n_{1},\ldots ,n_{k})\in \mathbb{Z}^{k}:d^{\ast }\biggl(\mathop{\bigcap }_{e_{1},\ldots ,e_{k}\in \{0,1\}}(E-(e_{1}n_{1}+\cdots +e_{k}n_{k}))\biggr)>d^{\ast }(E)^{2^{k}}-\unicode[STIX]{x1D700}\biggr\} & & \displaystyle \nonumber\end{eqnarray}$$is an$\text{AVIP}_{0}^{\ast }$-set. We then generalize this result to the case of ‘polynomial cubic configurations’$e_{1}p_{1}(n)+\cdots +e_{k}p_{k}(n)$, where the polynomials$p_{i}:\mathbb{Z}^{d}\longrightarrow \mathbb{Z}$are assumed to be sufficiently algebraically independent.

scholarly journals On a Sumset Conjecture of Erdős

2015 ◽  
Vol 67 (4) ◽  
pp. 795-809 ◽  
Author(s):  
Mauro Di Nasso ◽  
Isaac Goldbring ◽  
Renling Jin ◽  
Steven Leth ◽  
Martino Lupini ◽  
...  
Keyword(s):  
Positive Solution ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Amenable Groups ◽  
Banach Density ◽  
Infinite Sets ◽  
Erdös Conjecture

AbstractErdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

scholarly journals Boxes, extended boxes and sets of positive upper density in the Euclidean space

2021 ◽  
pp. 1-21
Author(s):  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ
Keyword(s):  
Euclidean Space ◽  
Arithmetic Progressions ◽  
Density Theorems ◽  
Higher Dimensional ◽  
Banach Density ◽  
Upper Density

Abstract We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2 n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

scholarly journals HIGH DENSITY PIECEWISE SYNDETICITY OF PRODUCT SETS IN AMENABLE GROUPS

2016 ◽  
Vol 81 (4) ◽  
pp. 1555-1562 ◽  
Author(s):  
MAURO DI NASSO ◽  
ISAAC GOLDBRING ◽  
RENLING JIN ◽  
STEVEN LETH ◽  
MARTINO LUPINI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Finite Subset ◽  
Amenable Group ◽  
High Density ◽  
Amenable Groups ◽  
Quantitative Version ◽  
Banach Density ◽  
Piecewise Syndetic ◽  
Product Sets

AbstractM. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤd, this result was first proven by the current set of authors using completely different techniques.

scholarly journals Kneser’s theorem for upper Banach density

2006 ◽  
Vol 18 (2) ◽  
pp. 323-343 ◽  
Author(s):  
Prerna Bihani ◽  
Renling Jin
Keyword(s):  
Banach Density

ON POINTS WITH POSITIVE DENSITY OF THE DIGIT SEQUENCE IN INFINITE ITERATED FUNCTION SYSTEMS

2016 ◽  
Vol 102 (3) ◽  
pp. 435-443
Author(s):  
ZHEN-LIANG ZHANG ◽  
CHUN-YUN CAO
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Positive Density ◽  
Digit Sequence ◽  
Iterated Function ◽  
Banach Density ◽  
Image Position

Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that $$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$ In this note, we investigate the size of the points whose digit sequences are strictly increasing and of upper Banach density one, which improves the work of Tong and Wang and Zhang and Cao.

Applications of Nonstandard Analysis in Additive Number Theory

2000 ◽  
Vol 6 (3) ◽  
pp. 331-341 ◽  
Author(s):  
Renling Jin
Keyword(s):  
Number Theory ◽  
Recent Progress ◽  
Nonstandard Analysis ◽  
Additive Number Theory ◽  
Additive Number ◽  
Banach Density

AbstractThis paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.

Product of Simplices and sets of positive upper density in ℝd

2017 ◽  
Vol 165 (1) ◽  
pp. 25-51 ◽  
Author(s):  
NEIL LYALL ◽  
ÁKOS MAGYAR
Keyword(s):  
Direct Proof ◽  
Two Dimensional ◽  
Banach Density ◽  
Upper Density ◽  
Product Of Simplices

AbstractWe establish that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4.We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δk1 and Δk2 are two fixed non-degenerate simplices of k1 + 1 and k2 + 1 points respectively, then any subset of ℝd of positive upper Banach density with d ⩾ k1 + k2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δk1 × Δk2.A new direct proof of the fact that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented.

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